{"id":13690,"date":"2022-04-15T09:59:16","date_gmt":"2022-04-15T13:59:16","guid":{"rendered":"https:\/\/www.purdue.edu\/freeform\/me274\/?p=13690"},"modified":"2024-10-11T07:13:14","modified_gmt":"2024-10-11T11:13:14","slug":"homework-h6-d","status":"publish","type":"page","link":"https:\/\/www.purdue.edu\/freeform\/me274\/chapter-6-discussions\/homework-h6-d\/","title":{"rendered":"Homework H6.B.10"},"content":{"rendered":"

\"\"<\/p>\n

Discussion and hints:<\/strong><\/em><\/p>\n

<\/p>\n

The derivation of the dynamical equation of motion (EOM) for a system is a straight-forward application of what we have learned from Chapter 5 in using the Newton-Euler equations. The goal in deriving the EOM is to end up with a single differential equation in terms of a single dependent variable that describes the motion of the system. Here in this problem, we want our EOM to be in terms of \u03b8<\/em>(t).<\/p>\n

Recall the following f<\/em>our-step plan<\/em><\/span> outline in the lecture book and discussed in lecture:<\/p>\n

Step 1: FBDs<\/strong><\/em>
\nDraw a FBD of the particle. It is recommended that you define and use a set of polar coordinates for this problem.<\/p>\n

Step 2: Kinetics (Newton)
\n<\/strong><\/em>Write down the Newton equation for the particle in the \u03b8<\/em>-direction.<\/p>\n

Step 3: Kinematics<\/strong><\/em>
\nDo you need any additional kinematics for this problem?<\/p>\n

Step 4: EOM<\/strong><\/em>
\nStep 2 should produce a single differential equation in terms of the dependent variable \u03b8<\/em>. Note that this EOM contains a nonlinear term of sin\u03b8. Recall that we can represent the sine function by its power series representation:\u00a0sin\u03b8<\/em> = \u03b8<\/em> – \u03b8<\/em>3<\/sup>\/3! + \u03b8<\/em>5<\/sup>\/5! – … \u00a0 \u00a0For small angles \u03b8, we see that this series could be approximated by its leading term, giving: sin\u03b8<\/em>\u00a0=\u00a0\u03b8<\/em>. The approximation for small angles of oscillation produces a LINEAR<\/em> differential equation. Use this approximation here.<\/p>\n

\n

Any questions?<\/p>\n","protected":false},"excerpt":{"rendered":"

Discussion and hints: The derivation of the dynamical equation of motion (EOM) for a system is a straight-forward application of what we have learned from Chapter 5 in using the Newton-Euler equations. The goal in deriving the EOM is to end up with a single differential equation in terms of a single dependent variable that … Continue reading Homework H6.B.10<\/span> →<\/span><\/a><\/p>\n","protected":false},"author":10,"featured_media":0,"parent":15017,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"class_list":["post-13690","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/www.purdue.edu\/freeform\/me274\/wp-json\/wp\/v2\/pages\/13690","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.purdue.edu\/freeform\/me274\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.purdue.edu\/freeform\/me274\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.purdue.edu\/freeform\/me274\/wp-json\/wp\/v2\/users\/10"}],"replies":[{"embeddable":true,"href":"https:\/\/www.purdue.edu\/freeform\/me274\/wp-json\/wp\/v2\/comments?post=13690"}],"version-history":[{"count":6,"href":"https:\/\/www.purdue.edu\/freeform\/me274\/wp-json\/wp\/v2\/pages\/13690\/revisions"}],"predecessor-version":[{"id":15189,"href":"https:\/\/www.purdue.edu\/freeform\/me274\/wp-json\/wp\/v2\/pages\/13690\/revisions\/15189"}],"up":[{"embeddable":true,"href":"https:\/\/www.purdue.edu\/freeform\/me274\/wp-json\/wp\/v2\/pages\/15017"}],"wp:attachment":[{"href":"https:\/\/www.purdue.edu\/freeform\/me274\/wp-json\/wp\/v2\/media?parent=13690"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}